Mastering Mathematics Smartly by Wee Wen Shih

A unique self-help website that provides comprehensive coverage of mathematics at A-level & beyond, written in a student-friendly style.

Vector product

Students typically find the topic of "Vectors" difficult.

Let us simplify its complexity by considering the applications of the vector product.

We use the vector product in these ways:

(1) To find the normal of a plane.

(2) To find the perpendicular distance of a point from a line.

(3) To find the area of a triangle or parallelogram.

See summary.

Always represent the problem description with any of the diagrams in the summary, to help you solve the question.

Lines and planes with given conditions

An application of the vector product is to find the equation of a line or a plane with given conditions. It is common for students to have some difficulty visualising how the resulting line or plane looks like.

Essentially, we tackle this type of problem by our understanding of the vector product, i.e. a x b is perpendicular to a and to b. Therefore, look closely at the given conditions and apply the definition of vector product.

Study this example to familiarise yourself with the solving approach.

Evaluating arbitrary vector products

Following our discussion on the evaluation of scalar products of arbitrary vectors, we continue with an example involving vector products.

Given the three position vectors \overrightarrow{OA} = \textbf{a}, \overrightarrow{OB} = \textbf{b}, \overrightarrow{OC} = \textbf{c} and a point D situated on the line BC, with \textbf{d} denoting the vector \overrightarrow{AD},
show that |\textbf{d}| = \frac{|(\textbf{b} - \textbf{a}) \times (\textbf{b} - \textbf{c})|}{|\textbf{b} - \textbf{c}|} = \frac{|(\textbf{c} - \textbf{a}) \times (\textbf{b} - \textbf{c})|}{|\textbf{b} - \textbf{c}|} .

Approach:
1. Draw a diagram (coming soon).

2. Observe that the vertical component of AC is equal to the vertical component of AB (details coming soon).